G(x) = min(f(t)) and g(x)=max(f(t))

If

f_{1}(x) = \left\{\begin{matrix} min \left \{ x^2, \left | x \right | \right \} & for \left | x \right |\leq 1\\ max \left \{ x^2, \left | x \right | \right \} & for \left | x \right |\ > 1 \end{matrix}\right.

f_{2}(x) = \left\{\begin{matrix} min \left \{ x^2, \left | x \right | \right \} & for \left | x \right |\ > 1\\ max \left \{ x^2, \left | x \right | \right \} & for \left | x \right |\leq 1 \end{matrix}\right.

and f(x)=f_{1}(x)-2f_{2}(x)

Then draw the graph of

g(x)=\left\{\begin{matrix} min\left \{ f(t) : -3\leqslant t\leq x, -3\leqslant x< 0 \right \}\\ max\left \{ f(t) : 0\leqslant t\leq x,0\leqslant x\leqslant 3\right \} \end{matrix}\right.

So lets draw the graph of $latex f_{1}(x) ,  $latex f_{2}(x) and g(x)

The graph of f(x)

Now let’s draw the graph of g(x)

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